Proposition 16.6.4.BBF7977859F1 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$ to $X$. For $\varphi ,\psi \in \mathrm{QPSH}(X)$, the following are equivalent:
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$\varphi \sim _{\mathcal{I}}\psi $;
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$\pi ^*\varphi \sim _{\mathcal{I}}\pi ^*\psi $.
Proof.
1 $\implies $ 2. This follows from proposition 16.6.2 4.
2 $\implies $ 1. This follows from the simple fact that
\[ \mathcal{I}(k\varphi )=\pi _* \left( \omega _{Y/X}\otimes \mathcal{I}(k\pi ^*\varphi ) \right),\quad \mathcal{I}(k\psi )=\pi _* \left( \omega _{Y/X}\otimes \mathcal{I}(k\pi ^*\psi ) \right) \]
for any $k\in \mathbb {Z}_{>0}$, where $\omega _{Y/X}$ is the relative dualizing sheaf. See [Proposition 5.8, Dem12].
$\square$
Pome
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